Question: Ben is 3 times as old as William. Twenty years ago, Ben was 7 times as old as William. How old is Ben now?
Answer: We can use the given information to write down two equations that describe the ages of Ben and William. Let Ben's current age be $b$ and William's current age be $w$ The information in the first sentence can be expressed in the following equation: $b = 3w$ Twenty years ago, Ben was $b - 20$ years old, and William was $w - 20$ years old. The information in the second sentence can be expressed in the following equation: $b - 20 = 7(w - 20)$ Now we have two independent equations, and we can solve for our two unknowns. Because we are looking for $b$ , it might be easiest to solve our first equation for $w$ and substitute it into our second equation. Solving our first equation for $w$ , we get: $w = b / 3$ . Substituting this into our second equation, we get: $b - 20 = 7($ $(b / 3)$ $- 20)$ which combines the information about $b$ from both of our original equations. Simplifying the right side of this equation, we get: $b - 20 = \dfrac{7}{3} b - 140$ Solving for $b$ , we get: $\dfrac{4}{3} b = 120$ $b = \dfrac{3}{4} \cdot 120 = 90$.